91Ó°ÊÓ

Use the trapezoidal rule and Simpson's rule with \(n=4,8, \ldots, 512\) to find approximate values of the area under the curve of \(y=f(x)\) for the following functions \(f(x)\) on the given intervals: (a) \(f(x)=e^{-x^{2}}, \quad 0 \leq x \leq 10\) (b) \(f(x)=\tan ^{-1}\left(1+x^{2}\right), \quad 0 \leq x \leq 2\) (c) \(\quad f(x)=\sqrt{x} e^{x}, \quad 0 \leq x \leq 1\)

(a) Consider using the trapezoidal rule \(T_{n}\) to estimate the integral $$ I=\int_{1}^{3} \log x d x $$ Give both a rigorous error bound for \(I-T_{n}\) and an asymptotic error estimate \(I-T_{n}\). Using the rigorous error bound, determine how large \(n\) should be in order that \(\left|I-T_{n}\right| \leq 5 \times 10^{-8}\). (b) Repeat with Simpson's rule.

Following is a table of values of the trapezoidal rule applied to the integral $$ I=\int_{0}^{1} \tan ^{-1} x d x=\frac{1}{4} \pi-\frac{1}{2} \ln 2=0.43882457311748 $$ Using the table, produce the Richardson's error estimate for \(T_{n}\) for \(n=16,32,64 .\) In addition, produce the corrected trapezoidal rule for \(n=64\). Using the true answer, given above, what is the error in yoar value for the corrected trapezoidal rule? \begin{tabular}{cc} \hline\(n\) & \(T_{n}\) \\ \hline 4 & \(0.4362066157\) \\ 8 & \(0.4381726803\) \\ 16 & \(0.4386617597\) \\ 32 & \(0.4387838797\) \\ 64 & \(0.4388149004\) \\ \hline \end{tabular}